Without knowing what the make/model of that faucet it, I'm going to say you don't need to get underneath that faucet.

The base of the faucet looks like an escutcheon that sits flush on the horizontal surface. Just above that seems to be a decorative cover (see pic), and looks like it might slide up. If it does, I think you'll see what needs to be done. I'm guessing the vertical shaft is the retaining screw into the cartridge and *may* be reverse thread?

Could be wrong on that.

I've been wrong before... it was on a Thursday.

If you're still interested in the make/model of the faucet, can you give us any more information about it? When was it installed? Any markings on any of the other parts?